Finite Group Databasebeta

← C3×#10604905 #14296922

Subgroups — C3×#10604905 #14296922

Order 1920.

1128 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×Dic10C211yesmaximal, characteristic
2C3×#65798C212yesmaximal
3C2×C3×#4005C214yesmaximal, complemented
4#10604905C311yesmaximal, characteristic, direct product factor, Hall
5C23×C3×C4×C5C411yescharacteristic
6C23×C3×Dic5C411yescharacteristic
7C23×C3×Dic5C2211yescharacteristic
8C22×C3×Dic10—24no—
9C22×C3×Dic10—28no—
10C22×C3×Dic10C412yes—
11C22×C3×Dic10C2212yes—
12C2×C3×#704C2214yes—
13C3×C22⋊#170—24no—
14C3×#4005—28no—
15C3×#1784—51nomaximal, Hall
16C23×Dic10C2×C311yescharacteristic
17#65798C2×C312yes—
18C2×#4005C2×C314yescomplemented
19C24×C3×C5C2×C411yescharacteristic
20C22×C3×C4×C5—22no—
21C22×C3×C4×C5—24no—
22C22×C3×C4×C5C2×C412yes—
23C22×C3×Dic5—22no—
24C22×C3×Dic5—24no—
25C22×C3×Dic5—24no—
26C22×C3×Dic5C2×C411yescharacteristic
27C22×C3×Dic5C2×C412yes—
28C22×C3×Dic5D414yes—
29C22×C3×Dic5C2311yescharacteristic
30C2×C3×Dic10—232no—
31C2×C3×Dic10—24no—
32C2×C3×Dic10—24no—
33C2×C3×Dic10—24no—
34C2×C3×Dic10—24no—
35C2×C3×Dic10—24no—
36C2×C3×Dic10D418yes—
37C3×#704—28no—
38C2×C3×#170—24no—
39C23×C3×Q8—51no—
40C2×C3×#101—54no—
41C3×#407—52no—
42C23×C4×C5C3×C411yescharacteristic
43C23×Dic5C3×C411yescharacteristic
44C23×Dic5C22×C311yescharacteristic
45C22×Dic10—24no—
46C22×Dic10—28no—
47C22×Dic10C3×C412yes—
48C22×Dic10C22×C312yes—
49C2×#704C22×C314yes—
50C22⋊#170—24no—
51#4005—28no—
52#1784—51noSylow
53C23×C3×C5—24no—
54C23×C3×C5—22no—
55C23×C3×C5C22⋊C412yes—
56C23×C3×C5C22×C411yescharacteristic
57C2×C3×C4×C5—28no—
58C2×C3×C4×C5—22no—
59C2×C3×C4×C5—22no—
60C2×C3×C4×C5C22⋊C414yes—
61C2×C3×Dic5—28no—
62C2×C3×Dic5—22no—
63C2×C3×Dic5—22no—
64C2×C3×Dic5—22no—
65C2×C3×Dic5—28no—
66C2×C3×Dic5C22⋊C412yes—
67C2×C3×Dic5C22⋊C414yes—
68C2×C3×Dic5C2×D412yes—
69C2×C3×Dic5C2×D414yes—
70C3×Dic10—216no—
71C3×Dic10—216no—
72C3×#170—44no—
73C22×C3×Q8—52no—
74C22×C3×Q8—52no—
75C22×C3×Q8—104no—
76C22×C3×Q8—108no—
77C23×C3×C4—51no—
78C23×C3×C4—51no—
79C23×C3×C4AGL(1,𝔽5)11yescharacteristic
80C2×C3×M16—54no—
81C3×#101—108no—
82C3×C22⋊C8—104no—
83C24×C5C2×C3×C411yescharacteristic
84C22×C4×C5—22no—
85C22×C4×C5—24no—
86C22×C4×C5C2×C3×C412yes—
87C22×Dic5—22no—
88C22×Dic5—24no—
89C22×Dic5—24no—
90C22×Dic5C2×C3×C411yescharacteristic
91C22×Dic5C2×C3×C412yes—
92C22×Dic5C3×D414yes—
93C22×Dic5C23×C311yescharacteristic
94C2×Dic10—232no—
95C2×Dic10—24no—
96C2×Dic10—24no—
97C2×Dic10—24no—
98C2×Dic10—24no—
99C2×Dic10—24no—
100C2×Dic10C3×D418yes—
101#704—28no—
102C2×#170—24no—
103C23×Q8—51no—
104C2×#101—54no—
105#407—52no—
106C22×C3×C5—28no—
107C22×C3×C5—22no—
108C22×C3×C5—22no—
109C22×C3×C5—22no—
110C22×C3×C5#10114yescomplemented
111C22×C3×C5C2×C22⋊C412yes—
112C22×C3×C5C2×C22⋊C411yescharacteristic
113C3×C4×C5—24no—
114C3×Dic5—22no—
115C3×Dic5—24no—
116C3×Dic5#12014yes—
117C24×C3C2×AGL(1,𝔽5)11yescharacteristic
118C2×C3×Q8—58no—
119C2×C3×Q8—104no—
120C2×C3×Q8—104no—
121C2×C3×Q8—104no—
122C2×C3×Q8—104no—
123C2×C3×Q8—1032no—
124C2×C3×Q8—104no—
125C22×C3×C4—22no—
126C22×C3×C4—24no—
127C22×C3×C4—51no—
128C22×C3×C4—54no—
129C22×C3×C4—52no—
130C22×C3×C4—51no—
131C22×C3×C4—104no—
132C22×C3×C4—104no—
133C22×C3×C4—102no—
134C22×C3×C4C2×AGL(1,𝔽5)12yes—
135C3×M16—108no—
136C2×C3×C8—104no—
137C23×C5—24no—
138C23×C5—22no—
139C23×C5C3×C22⋊C412yes—
140C23×C5C22×C3×C411yescharacteristic
141C2×C4×C5—28no—
142C2×C4×C5—22no—
143C2×C4×C5—22no—
144C2×C4×C5C3×C22⋊C414yes—
145C2×Dic5—28no—
146C2×Dic5—22no—
147C2×Dic5—22no—
148C2×Dic5—22no—
149C2×Dic5—28no—
150C2×Dic5C3×C22⋊C412yes—
151C2×Dic5C3×C22⋊C414yes—
152C2×Dic5C2×C3×D412yes—
153C2×Dic5C2×C3×D414yes—
154Dic10—216no—
155Dic10—216no—
156#170—44no—
157C22×Q8—52no—
158C22×Q8—52no—
159C22×Q8—104no—
160C22×Q8—108no—
161C23×C4—51no—
162C23×C4—51no—
163C23×C4C3×AGL(1,𝔽5)11yescharacteristic
164C2×M16—54no—
165#101—108no—
166C22⋊C8—104no—
167C2×C3×C5—24no—
168C2×C3×C5—22no—
169C2×C3×C5#37911yescharacteristic
170C2×C3×C5C2×#10112yes—
171C23×C3—22no—
172C23×C3—24no—
173C23×C3C22⋊AGL(1,𝔽5)12yes—
174C23×C3C22×AGL(1,𝔽5)11yescharacteristic
175C3×Q8—1016no—
176C3×Q8—1016no—
177C2×C3×C4—28no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—54no—
181C2×C3×C4—52no—
182C2×C3×C4—54no—
183C2×C3×C4—52no—
184C2×C3×C4—108no—
185C2×C3×C4—108no—
186C2×C3×C4—102no—
187C2×C3×C4—102no—
188C2×C3×C4—102no—
189C2×C3×C4C22⋊AGL(1,𝔽5)14yes—
190C3×C8—204no—
191C22×C5—28no—
192C22×C5—22no—
193C22×C5—22no—
194C22×C5—22no—
195C22×C5C3×#10114yescomplemented
196C22×C5C2×C3×C22⋊C412yes—
197C22×C5C2×C3×C22⋊C411yescharacteristic
198C4×C5—24no—
199Dic5—22no—
200Dic5—24no—
201Dic5C3×#12014yes—
202C24C2×C3×AGL(1,𝔽5)11yescharacteristic
203C2×Q8—58no—
204C2×Q8—104no—
205C2×Q8—104no—
206C2×Q8—104no—
207C2×Q8—104no—
208C2×Q8—1032no—
209C2×Q8—104no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—51no—
213C22×C4—54no—
214C22×C4—51no—
215C22×C4—52no—
216C22×C4—104no—
217C22×C4—104no—
218C22×C4—102no—
219C22×C4C2×C3×AGL(1,𝔽5)12yes—
220M16—108no—
221C2×C8—104no—
222C3×C5#178411yescharacteristic, complemented, Hall
223C22×C3—28no—
224C22×C3—22no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3#400514yescomplemented
228C22×C3C2×C22⋊AGL(1,𝔽5)12yes—
229C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
230C3×C4—24no—
231C3×C4—54no—
232C3×C4—104no—
233C3×C4—102no—
234C2×C5—24no—
235C2×C5—22no—
236C2×C5C3×#37911yescharacteristic
237C2×C5C2×C3×#10112yes—
238C23—24no—
239C23—22no—
240C23C3×C22⋊AGL(1,𝔽5)12yes—
241C23C22×C3×AGL(1,𝔽5)11yescharacteristic
242Q8—1016no—
243Q8—1016no—
244C2×C4—28no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—52no—
248C2×C4—54no—
249C2×C4—52no—
250C2×C4—54no—
251C2×C4—108no—
252C2×C4—108no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—102no—
256C2×C4C3×C22⋊AGL(1,𝔽5)14yes—
257C8—204no—
258C2×C3—22no—
259C2×C3—24no—
260C2×C3C2×#400512yes—
261C2×C3#6580011yescharacteristic
262C5C3×#178411yescharacteristic, complemented, Sylow
263C22—28no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22C3×#400514yescomplemented
268C22C2×C3×C22⋊AGL(1,𝔽5)12yes—
269C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
270C4—24no—
271C4—54no—
272C4—102no—
273C4—104no—
274C3#1060490511yescharacteristic, direct product factor, Sylow
275C2—24no—
276C2—22no—
277C2C2×C3×#400512yes—
278C2C3×#6580011yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC23×C3×C4×C5C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC24×C3×C5C2×C4
cosocleC22×Dic5C23×C3
hypercenterC23×C3×C4AGL(1,𝔽5)
hypocenterC5C3×#1784
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×C3×#4005C2semidirect
4. #10604905C3direct (N × H)
18. C2×#4005C2×C3semidirect
110. C22×C3×C5#101semidirect
195. C22×C5C3×#101semidirect
222. C3×C5#1784semidirect
227. C22×C3#4005semidirect
262. C5C3×#1784semidirect
267. C22C3×#4005semidirect
274. C3#10604905direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×C3×Dic102
C3×#657982
C2×C3×#40052
#106049053
C3×#17845